cp's OEIS Frontend

A214549 Decimal expansion of 4*Pi^2/27.

%I A214549 #49 Nov 03 2023 05:26:35
%S A214549 1,4,6,2,1,6,3,6,1,4,9,7,6,2,0,1,2,7,6,8,6,4,3,6,9,0,3,7,0,1,8,6,8,9,
%T A214549 0,5,7,0,8,3,5,1,1,0,2,3,2,9,4,9,3,1,9,4,4,6,5,3,8,2,9,5,3,7,2,1,7,7,
%U A214549 8,4,4,1,8,1,3,6,1,7,8,5,5,4,5,1,8,7,8,1,2,4,4,9,9
%N A214549 Decimal expansion of 4*Pi^2/27.
%C A214549 Represents the value of the Dirichlet series for A011655 (principal Dirichlet character mod 3) at s=2.
%C A214549 Equals the asymptotic mean of the abundancy index of the numbers that are not divisible by 3 (A001651). - _Amiram Eldar_, May 12 2023
%H A214549 G. C. Greubel, <a href="/A214549/b214549.txt">Table of n, a(n) for n = 1..10000</a>
%H A214549 R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-Series</a>, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.
%H A214549 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A214549 Equals (4/3)*A100044.
%F A214549 Equals Sum_{n>=0} (1/(3*n+1)^2 + 1/(3*n+2)^2).
%F A214549 From _Peter Luschny_, May 13 2020: (Start)
%F A214549 Equals (8/9) * Sum_(k>=1) 1/k^2 =8/9 *A013661.
%F A214549 Equals -(16/9) * Sum_(k>=1) (-1)^k/k^2 = -16/9 * A072691.
%F A214549 Equals (64/27) * ( Integral_{x=0..1} sqrt(1 - x^2) )^2 = 64/27 * A091476. (End)
%F A214549 Equals Integral_{x=0..oo} log(x)/(x^3 - 1) dx. - _Amiram Eldar_, Aug 12 2020
%e A214549 1.4621636149762012768643690370186...
%p A214549 evalf(4*Pi^2/27) ;
%t A214549 RealDigits[(4Pi^2)/27,10,120][[1]] (* _Harvey P. Dale_, Dec 20 2012 *)
%o A214549 (PARI) 4*Pi^2/27 \\ _G. C. Greubel_, Mar 08 2018
%o A214549 (Magma) R:= RealField(); 4*Pi(R)^2/27; // _G. C. Greubel_, Mar 08 2018
%o A214549 (Magma) R:=RealField(106); SetDefaultRealField(R); n:=4*Pi(R)^2/27; Reverse(Intseq(Floor(10^105*n))); // _Bruno Berselli_, Mar 13 2018
%o A214549 (Julia)
%o A214549 using Nemo
%o A214549 R = RealField(310)
%o A214549 t = const_pi(RR) + const_pi(RR); s = t * t
%o A214549 s / RR(27) |> println # _Peter Luschny_, Mar 13 2018
%Y A214549 Cf. A001651, A011655, A100044.
%K A214549 nonn,cons,easy
%O A214549 1,2
%A A214549 _R. J. Mathar_, Jul 20 2012